Practice question · Multiple choice
An eigenvector is described as a direction the matrix cannot bend. What makes those directions worth finding?
Hints
- Apply the matrix ten times to an eigenvector. What do you get?
- Ask what the hard part of matrix powers is, and whether it survives in an eigen-direction.
Show the answer
D. Along them the matrix acts as a single number
Why
Aᵏv = λᵏv, so a hundred steps of a transition process is one exponentiation rather than a hundred matrix products. Finding a coordinate system where a complicated operator becomes multiplication is the recurring move of linear algebra, and it is what diagonalisation formalises.
Practise Eigenvalues and Eigenvectors
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